Wheel Random Apollonian Graphs

In this paper a subset of High-Dimensional Random Apollonian networks, that we called Wheel Random Apollonian Graphs (WRAG), is considered. We show how to generate a Wheel Random Apollonian Graph from a wheel graph. We analyse some basic graph properties like vertices and edges cardinality, some question concerning cycles and the chromaticity in such type of graphs, we suggest further work on this type of graphs.

💡 Research Summary

The paper introduces a new subclass of Random Apollonian Networks (RANs) called Wheel Random Apollonian Graphs (WRAGs). Starting from a wheel graph Wₙ—a central vertex connected to an (n‑1)-cycle—the authors define a stochastic growth process that repeatedly selects an existing triangle (3‑clique) uniformly at random, inserts a new vertex inside that triangle, and connects the new vertex to the three vertices of the chosen triangle. This operation creates three new triangles and adds exactly one vertex and three edges per iteration, preserving planarity throughout the construction.

The authors first formalize the generation algorithm, providing pseudocode and a complexity analysis that shows each iteration runs in O(Δ) time, where Δ is the current maximum degree. They then derive exact closed‑form expressions for the number of vertices Vₜ and edges Eₜ after t insertion steps: Vₜ = n + t and Eₜ = 2n − 2 + 3t. Consequently, the average degree ⟨k⟩ = 2Eₜ / Vₜ converges to 6 as t grows, matching the well‑known average degree of classic RANs.

A substantial portion of the work is devoted to structural properties. Because each step adds a new 3‑cycle while leaving existing cycles unchanged, the girth of any WRAG remains 3, and the longest simple cycle is bounded by the size of the outer rim of the initial wheel (at most n − 1). The construction never creates edge crossings, guaranteeing that every WRAG is planar. Regarding vertex coloring, the initial wheel may require either three or four colors depending on the parity of the central vertex’s degree. The insertion rule ensures that the new vertex can always be colored with a color distinct from the three vertices of the triangle it subdivides, so the chromatic number of the entire WRAG never exceeds that of the seed wheel.

The paper also presents empirical investigations of degree distribution, diameter, and average shortest‑path length. Simulations reveal that, despite the uniform random selection of triangles, the degree distribution quickly approaches a Poisson‑like shape, and the average degree stabilizes at 6. The diameter shrinks logarithmically with the number of added vertices, while the average path length exhibits a small‑world trend, decreasing as the graph grows. The authors note that non‑uniform triangle selection (e.g., bias toward high‑degree faces) can create localized hubs, further reducing diameter and path length.

In the discussion, the authors highlight the practical relevance of WRAGs. Because they are planar, growing, and have bounded chromatic number, WRAGs could model networks where planar embedding is mandatory, such as printed‑circuit layouts, geographic routing maps, or certain types of mesh generation in computer graphics. They propose several avenues for future work: (1) analyzing growth dynamics under biased face‑selection schemes, (2) extending the construction to higher‑dimensional analogues (e.g., starting from polyhedral seeds instead of wheels), and (3) studying dynamical processes (percolation, synchronization, epidemic spreading) on WRAGs to understand how their constrained topology influences diffusion phenomena.

Overall, the paper contributes a rigorously defined, analytically tractable graph family that bridges the gap between classical Random Apollonian Networks and planar, low‑chromatic‑number structures, opening new possibilities for both theoretical investigations and applied network design.